F. Labourie [arXiv:1212.5015] characterized the Hitchin components for PSL(n,R) for any n>1 by using the swapping algebra, where the swapping algebra should be understood as a ring equipped with a Poisson bracket. We introduce the rank n swapping algebra, which is the quotient of the swapping algebra by the (n+1)×(n+1) determinant relations. The main results are the well-definedness of the rank n swapping algebra and the "cross-ratio" in its fraction algebra. As a consequence, we use the sub fraction algebra of the rank n swapping algebra generated by these "cross-ratios" to characterize the PSL(n,R) Hitchin component for a fixed n>1. We also show the relation between the rank 2 swapping algebra and the cluster X space.

An analogue of the well-known $$ \frac{3}{{16}} $$ lower bound for the first eigenvalue of the Laplacian for a congruence hyperbolic surface is proven for a congruence tower associated with any non-elementary subgroup$L$of SL(2,$Z$). The proof in the case that the Hausdorff of the limit set of$L$is bigger than $$ \frac{1}{2} $$ is based on a general result which allows one to transfer such bounds from a combinatorial version to this archimedian setting. In the case that delta is less than $$ \frac{1}{2} $$ we formulate and prove a somewhat weaker version of this phenomenon in terms of poles of the corresponding dynamical zeta function. These “spectral gaps” are then applied to sieving problems on orbits of such groups.

No abstract uploaded!

Let $k$ be a field and $k(x_0,\ldots,x_{p-1})$ be the rational function field of $p$ variables over $k$ where $p$ is a prime number. Suppose that $G=\langle\sigma\rangle \simeq C_p$ acts on $k(x_0,\ldots,x_{p-1})$ by $k$-automorphisms defined as $\sigma:x_0\mapsto x_1\mapsto\cdots\mapsto x_{p-1}\mapsto x_0$. Denote by $P$ the set of all prime numbers and define $P_0=\{p\in P:\bm{Q}(\zeta_{p-1})$ is of class number one$\}$ where $\zeta_n$ a primitive $n$-th root of unity in $\bm{C}$ for a positive integer $n$; $P_0$ is a finite set by \cite{MM}. Theorem. Let $k$ be an algebraic number field and $P_k=\{p\in P: p$ is ramified in $k\}$. Then $k(x_0,\ldots,x_{p-1})^G$ is not stably rational over $k$ for all $p\in P\backslash (P_0\cup P_k)$.

No abstract uploaded!